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Theorem sseqtrrd 3136
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)
Hypotheses
Ref Expression
sseqtrrd.1  |-  ( ph  ->  A  C_  B )
sseqtrrd.2  |-  ( ph  ->  C  =  B )
Assertion
Ref Expression
sseqtrrd  |-  ( ph  ->  A  C_  C )

Proof of Theorem sseqtrrd
StepHypRef Expression
1 sseqtrrd.1 . 2  |-  ( ph  ->  A  C_  B )
2 sseqtrrd.2 . . 3  |-  ( ph  ->  C  =  B )
32eqcomd 2145 . 2  |-  ( ph  ->  B  =  C )
41, 3sseqtrd 3135 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    C_ wss 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084
This theorem is referenced by:  sseqtrrid  3148  fnfvima  5652  tfrlemiubacc  6227  tfr1onlemubacc  6243  tfrcllemubacc  6256  rdgivallem  6278  nnnninf  7023  dfphi2  11907  ctinf  11954  toponss  12207  ssntr  12305  iscnp3  12386  cnprcl2k  12389  tgcn  12391  tgcnp  12392  ssidcn  12393  cncnp  12413  txcnp  12454  imasnopn  12482  hmeontr  12496  blssec  12621  blssopn  12668  xmettx  12693  metcnp  12695  nnsf  13285  nninfsellemsuc  13294
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